“We the Jury ...”(Conditional Probability in Criminal Justice)

For example, suppose you are on the jury in a murder trial. The
prosecution introduces DNA evidence from the scene, and establishes
with expert testimony that “the odds are a billion to 1 against anyone
else sharing this DNA.”

You might think that means the odds are a billion to 1 against the
defendant, or the probability is a negligible 1/1,000,000,000 that
he’s innocent. But actually that’s the probability that he shares this
DNA if (given that) he’s innocent, not the probability that
he’s innocent if he shares this DNA.

In fact, if the odds are a billion to 1 against sharing this DNA,
then there are likely to be five other people out of the world’s
population of 6 billion who do share this DNA. Since five of the
six are certainly innocent, the defendant’s odds of being innocent
if he shares this DNA are 5 in 6, not 1 in a billion.

You might object, “But those other five people could be on the
other side of the planet when the murder happened!” That’s a good
thing to wonder about, and the defense would certainly have to address
that. In the meantime, here’s an example that doesn’t have that
problem, from page 76 of John Allen Paulos’s book Once upon a
Number:

Imagine a city of approximately one million people, a heinous
murder has been committed, and the only evidence available indicates
that the murderer has a very rare sort of mustache. [You could
pick any other physical characteristic if “mustache” seems far fetched
to you — a distinctive scar, perhaps, or extreme height.
—SB]

Assume
further that only two residents of the city have such mustaches. One
of these people is innocent, the other guilty. Then the probability
that an innocent person has this rare form of mustache is one in a
million; one out of the one million people has such a mustache. By
contrast, the probability that a person having such a mustache is
innocent is one in two! [Two people have such mustaches,
one of them is innocent.] Circumstantial evidence, motive, and
further physical evidence therefore should always be sought to bolster
any single piece of forensic evidence.